Essential dimension of double covers of symmetric and alternating groups

2019/06/09 by Zinovy Reichstein, Reichstein, Zinovy, Abhishek Shukla +1
Mathematics · #11E04 #20B30 #20C23 #20G10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1906.03698

openalex publication_date 2019/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

I. Schur studied double covers \widetilde\Sym^±_n and \widetilde\Alt_n of symmetric groups \Sym_n and alternating groups \Alt_n, respectively. Representations of these groups are closely related to projective representations of \Sym_n and \Alt_n; there is also a close relationship between these groups and spinor groups. We study the essential dimension \ed(\widetilde\Sym^±_n) and \ed(\widetilde\Alt_n). We show that over a base field of characteristic ≠ 2, \ed(\widetilde\Sym^±_n) and \ed(\widetilde\Alt_n) grow exponentially with n, similar to \ed(\Spin_n). On the other case, in characteristic 2, they grow sublinearly, similar to \ed(\Sym_n) and \ed(\Alt_n). We give an application of our result in good characteristic to the theory of trace forms.

Related

• Essential dimension of symmetric groups in prime characteristic
• Essential dimensions of finite groups
• Representations of symmetric and alternating groups and their double covers that remain irreducible modulo every prime
• Essential dimension of the spin groups in characteristic 2
• Modular functions and resolvent problems